t0_from_tminus1_to_t0_bridge
plain-language theorem explainer
Given a T-1→T0 bridge certificate (Boolean absolute floor plus its normalized two-point cost), the T0 surface follows: logic is the zero/positive split of recognition work. Anyone citing the complete inevitability chain from absolute floor through T0–T8 needs this edge. The proof is a structure constructor that reads the bridge’s normalized floor fields and the Boolean cost’s additivity.
Claim. If $B$ is a T-1$\to$T0 bridge certificate (Boolean absolute-floor witness, canonical two-point normalization, floor configuration, and unit-normalized Boolean recognition-work cost), then logic is forced: the Boolean floor carries a recognition-work cost with consistent configurations at cost $0$, inconsistent ones at positive cost, zero cost iff consistency, and independent additivity.
background
The Unified Forcing Chain module claims every level T-1 through T8 is forced from the cost foundation (Recognition Composition Law plus normalization and calibration). T-1 is the absolute floor: a meta-language Prop distinction in a non-singleton universe. T0 is the claim that logic is not pre-given but emerges as the zero/positive split of recognition work on that floor.
The bridge certificate packages four pieces: a Boolean absolute-floor witness, its canonical two-point normalization, the extracted Boolean configuration interface, and the unit-normalized Boolean recognition-work cost. The T0 surface then asks for a recognition-work constraint on Bool, cheap consistency (C(false)=0), expensive contradiction (positive cost on inconsistent states), emergent logic (zero iff consistent), and independent additivity of the Boolean cost.
This edge is the non-vacuous link missing from older aggregates: absolute floor supplies Boolean distinction; that distinction carries a concrete cost with dichotomy and additivity; Logic-from-Cost is reached through that interface.
proof idea
Term-mode structure construction, not a tactic script. Recognition work is taken from the bridge’s normalized floor. Consistency-cheap is the normalized floor cost’s zero-on-empty fact. Contradiction-expensive is the reverse direction of the floor cost’s positive-iff-inconsistent equivalence, applied pointwise. Logic-emergent is the floor cost’s zero-iff-consistent equivalence. Additive independence is the Boolean recognition cost’s additivity lemma on the T-1→T0 path. No further algebraic work; the bridge already holds the normalized payload.
why it matters
This is the explicit T-1→T0 edge in the complete inevitability chain. Downstream, the audit equality equates the direct global T0 surface with this routed construction. T1 (Meta-Principle) is obtained as a corollary of this routed T0; T2 (discreteness) and T3 (ledger) thread the same bridge through successive corollaries. Without it, T0 would sit as an assumed logic layer rather than a cost-forced surface beneath the Meta-Principle. Framework landmark: T0 in the forcing chain (logic from cost minimization, consistency cheap). It closes the gap between Absolute Floor Closure and the rest of T0–T8.
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